Theorem cvlatcvr1 29836

Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
cvlatcvr1.j	|- .\/ = ( join ` K )
cvlatcvr1.c	|- C = ( <o ` K )
cvlatcvr1.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
cvlatcvr1	|- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Proof of Theorem cvlatcvr1

Step	Hyp	Ref	Expression
1		simp13 989	. . . 4 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. CvLat )
2		cvlatl 29820	. . . 4 |- ( K e. CvLat -> K e. AtLat )
3	1, 2	syl 16	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. AtLat )
4		eqid 2412	. . . 4 |- ( meet ` K ) = ( meet ` K )
5		eqid 2412	. . . 4 |- ( 0. ` K ) = ( 0. ` K )
6		cvlatcvr1.a	. . . 4 |- A = ( Atoms ` K )
7	4, 5, 6	atnem0 29813	. . 3 |- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
8	3, 7	syld3an1 1230	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
9		eqid 2412	. . . 4 |- ( Base ` K ) = ( Base ` K )
10	9, 6	atbase 29784	. . 3 |- ( P e. A -> P e. ( Base ` K ) )
11		cvlatcvr1.j	. . . 4 |- .\/ = ( join ` K )
12		cvlatcvr1.c	. . . 4 |- C = ( <o ` K )
13	9, 11, 4, 5, 12, 6	cvlcvrp 29835	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. ( Base ` K ) /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
14	10, 13	syl3an2 1218	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
15	8, 14	bitrd 245	1 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ w3a 936   = wceq 1649   e. wcel 1721   =/= wne 2575   class class class wbr 4180   ` cfv 5421  (class class class)co 6048   Basecbs 13432   joincjn 14364   meetcmee 14365   0.cp0 14429   CLatccla 14499   OMLcoml 29670   <o ccvr 29757   Atomscatm 29758   AtLatcal 29759   CvLatclc 29760

This theorem is referenced by:  cvlatcvr2  29837  atcvr1  29911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-undef 6510  df-riota 6516  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-lat 14438  df-clat 14500  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817